The existence of partitioned generalized balanced tournament designs with block size 3
type: Item
instance of: scholarly article
title: The existence of partitioned generalized balanced tournament designs with block size 3
zbMATH Open document ID: 0869.05012
DOI: 10.1023/A:1008250908638
author: Esther R. Lamken
published in: Designs, Codes and Cryptography
publication date: 1997-08-27T00:00:00Z
review text: Let \(V\) be a set with \(kn\) elements. If the blocks of a \(\text{BIBD}(kn,k,k-1)\) defined on \(V\) can be arranged into an \(n\times(nk-1)\) array such that (i) each element in \(V\) occurs precisely in one cell of each column, and (ii) every element in \(V\) is contained in at most \(k\) cells of each row, then the configuration so obtained is called a generalized balanced tournament design, denoted by \(\text{GBTD}(n,k)\). Suppose, further, we can partition the columns of GBTD into \((k+1)\) sets \(B_1,B_2,\dots,B_{k+1}\) where \(|B_i|= n\) for \(i=1,2,3,\dots,k-2\), \(|B_i|= n-1\) for \(i=k-1, k\) and \(|B_i|=1\) for \(i=k+1\) such that (i) each element of \(V\) appears exactly once in each row and column of \(B_i\) \((i=1,2,\dots,k-2)\), and (ii) every element of \(V\) occurs exactly once in each row and column of \(B_i\cup B_{i+1}\) for \(i=k-1, k\), then GBTD is called partitioned, denoted by \(\text{PGBTD}(n,k)\). The author proves the existence of \(\text{PGBTD}(n,k)\) for \(k=3\) with the possible exceptions of 165 values of \(n\). These results are then used to establish the existence of a class of Kirkman squares in diagonal form. Finally, an appendix containing tables is given which describe the constructions for \(\text{PGBTD}(n',3)\) with \(461\leq n'\leq 1745\) and \(n'\equiv i\pmod 9\) for \(i=2,3,\dots,8\). A computer program was used to generate these tables.
Mathematics Subject Classification ID: 05B05
zbMATH DE Number: 1002297
zbMATH Keywords: generalized balanced tournament design
zbMATH Keywords: Kirkman squares
reviewed by: D. V. Chopra
MaRDI profile type: Publication
full work available at URL: a:1008250908638
OpenAlex ID: W1585607283
description: scientific article; zbMATH DE number 1002297